Regularity of solutions to an integral equation associated with Bessel potential. (English) Zbl 1237.45002

Han, Xiaolong; Lu, Guozhen

Commun. Pure Appl. Anal. 10, No. 4, 1111-1119 (2011).

The authors study the differential equation in \(\mathbb{R}^n\) \[ (I-\Delta)^{\frac{\alpha}{2}} u = u^\beta,\quad \alpha >0, \;\beta > 1. \] They prove that if \(u\) is a positive solution of this equation in the space \(L^q (\mathbb{R}^n), \;q > \max (\beta, \frac{n(\beta - 1)}{\alpha}),\) then \(u\) is a bounded and Lipschitz continuous function.
There is a survey of previous results in the article.

Reviewer: Anatoly Filip Grishin (Khar’kov)

MSC:

45E10	Integral equations of the convolution type (Abel, Picard, Toeplitz and Wiener-Hopf type)
35J60	Nonlinear elliptic equations
45M20	Positive solutions of integral equations

Keywords:

integral equation; Bessel potential; regularity lifting; \(L^\infty\) estimate; Lipschitz continuity estimate; differential equation; positive solution